GCF for 8, 21 and 75

"Greatest Common Factor of 3 Numbers" Calculator
Greatest Common Factor of
,
and

What's the Greatest Common Factor (GCF) of 8, 21 and 75?

Answer

GCF of 8, 21 and 75 is
1

(One)

1 is the largest positive integer that divides 8, 21 and 75 without a remainder — every other common divisor of the three divides it too.

Finding the GCF of 8, 21 and 75 by Prime Factorization

Split all three numbers into prime factors, then keep every prime that appears in all three of them, each taken as many times as it appears in the poorest factorization. Multiplying what is left gives the greatest common factor:

All Prime Factors of 8: 2 × 2 × 2 (exponent form: 23)

All Prime Factors of 21: 3 × 7 (exponent form: 3 × 7)

All Prime Factors of 75: 3 × 5 × 5 (exponent form: 3 × 52)

Prime factor82175Lowest power
223———
3—33—
5——52—
7—7——

Not one prime factor is shared by all three numbers, which makes 8, 21 and 75 relatively prime as a set. The only positive integer dividing all of them is 1.

Checking the GCF of 8, 21 and 75 Two Numbers at a Time

The greatest common factor is associative, so three numbers can be handled as two pairs — and this is what makes the Euclidean algorithm usable here: run it on the first pair, then run it again on that result and the third number.

GCF(a, b, c) = GCF(GCF(a, b), c)

GCF(8, 21) = 1

GCF(1, 75) = 1

GCF(8, 21, 75) = 1

The answer checks out by division — all three quotients come out whole:

8 ÷ 1 = 8

21 ÷ 1 = 21

75 ÷ 1 = 75

8, 21 and 75 have no common factor left — that is what makes 1 the greatest one.

Related Calculations

See Also

About "Greatest Common Factor of 3 Numbers" Calculator

This calculator finds the Greatest Common Factor (GCF) of three numbers at once, without doing it in two passes. For example, it can help you find out what's the Greatest Common Factor (GCF) of 8, 21 and 75? (The answer is: 1). Enter the first number (e.g. '8'), the second (e.g. '21') and the third (e.g. '75'), then hit the 'Calculate' button.
The Greatest Common Factor of three numbers — also called the greatest common divisor (GCD) or highest common factor (HCF) — is the largest positive integer that divides all three of them with zero remainder
"Greatest Common Factor of 3 Numbers" Calculator
Greatest Common Factor of
,
and

GCF Table

Number 1Number 2Number 3GCF
821601
821611
821621
821631
821641
821651
821661
821671
821681
821691
821701
821711
821721
821731
821741
821751
821761
821771
821781
821791
821801
821811
821821
821831
821841
821851
821861
821871
821881
821891

FAQ

What's the Greatest Common Factor (GCF) of 8, 21 and 75?

GCF of 8, 21 and 75 is 1

What prime factors do 8, 21 and 75 share?

None at all: 8, 21 and 75 have no prime factor in common, so the only positive integer dividing all three is 1 and their GCF is 1.

Can you get the GCF of 8, 21 and 75 two numbers at a time?

Yes, and it is the check this page prints. The GCF of 8 and 21 is 1, and the GCF of 1 and 75 is 1 — the same answer, because the greatest common factor is associative and does not depend on the order of the three numbers.

How does the GCF simplify the ratio 8 : 21 : 75?

Divide every term by the GCF 1: 8 ÷ 1 = 8, 21 ÷ 1 = 21 and 75 ÷ 1 = 75. So 8 : 21 : 75 in lowest terms is 8 : 21 : 75, and no whole number other than 1 divides all three of those.

See Also